i took arbitrary elements from the conjugacy classes represented by , viz,
these two should commute given that commute.
when i work it out it comes down to saying that commute iff
Calling we have that should commute. here i am stuck.

Hmm.. well that's not the way to go. Just because representatives of classes commute, doesn't mean that all elements between the classes commute. For example, consider the classes . There must be another way..

EDIT: Try this.

First, we need the fact that, for any finite group and subgroup , .

Now, we let act on itself by conjugation. Since it is assumed that the commute pairwise, we have for all . That is, for each , (the stabilizer of under this group action).

Let be arbitrary. Then there exist such that . Hence for each .

We have shown that, for any , we can find so that . It follows that (for each ).

By the claim above, since is finite, this forces for each . That is, the conjugacy class of is . We know that this means and, since these elements form a complete set of representatives for the conjugacy classes of , , and the group is abelian.

Hmm.. well that's not the way to go. Just because representatives of classes commute, doesn't mean that all elements between the classes commute. For example, consider the classes . There must be another way..

EDIT: Try this.

First, we need the fact that, for any finite group and subgroup , .

Now, we let act on itself by conjugation. Since it is assumed that the commute pairwise, we have for all . That is, for each , (the stabilizer of under this group action).

Let be arbitrary. Then there exist such that . Hence for each .

We have shown that, for any , we can find so that . It follows that (for each ).

By the claim above, since is finite, this forces for each . That is, the conjugacy class of is . We know that this means and, since these elements form a complete set of representatives for the conjugacy classes of , , and the group is abelian.

now that's quite something!
i agree, the proof is complete.
but i can understand the proof. there was no way that i would think that way.
how does all that come to ones mind?

EDIT: AND THIS TOO IS A SERIOUS QUESTION, AS SERIOUS AS THE THREAD ITSELF!!

Last edited by abhishekkgp; Mar 7th 2011 at 07:15 PM.
Reason: left out something important.